After two decades of routing well-heeled travelers out of Louis Armstrong International, I can tell you that no itinerary tests patience quite like the journey to Bosnia's mountain-ringed capital. There is no nonstop option, and there never has been: business class from New Orleans to Sarajevo requires a minimum of two connections, typically funneled through Atlanta or JFK before crossing the Atlantic to Frankfurt, Munich, or Istanbul. That routing reality is the single most important fact for planners—the total journey clocks in well beyond the advertised ~10-hour air time once you account for layovers, often stretching to 16-20 hours door to door.
My preferred build is Delta's domestic leg into JFK or Atlanta, connecting onto Lufthansa's transatlantic metal, ideally one of the reconfigured aircraft featuring the new Allegris business suite with its sliding door and adjustable panels—a meaningful upgrade over the older staggered seats still flying on some Munich and Frankfurt rotations, so it's worth checking your aircraft type at booking. From either German hub, the final hop to Sarajevo is short-haul intra-European product, so don't expect lie-flat seats on that segment—stretch and eat well during the long-haul portion instead. Turkish Airlines via Istanbul is my dark-horse recommendation: their business class lounge at IST is genuinely one of the best in the world, with a cinema room and live cooking stations, making a long layover feel like a destination unto itself rather than dead time.
Sarajevo Airport itself is compact and efficient, so arrival logistics are painless—budget extra buffer time instead for winter fog delays, which occasionally snarl the final approach into SJJ. Seasonally, January through March and October through November consistently offer the softest business class fares, since summer brings diaspora travelers home in heavy volume and prices firm up accordingly. Given the complexity of this three-airline puzzle, I'd strongly suggest setting up automated fare alerts rather than manually checking—award space and premium cabin pricing on this route shift often enough that a tracked alert will catch windows you'd otherwise miss entirely.